Calculus 3: Tangent Planes and Linear Approximation (Video #14) | Math with Professor V

This paragraph introduces the concept of tangent planes and linear approximations in the context of calculus for functions of multiple variables. It compares the role of tangent lines for single-variable functions to tangent planes for surfaces. The paragraph explains how tangent planes provide close approximations of function values near a point of tangency, similar to how tangent lines do for curves. It sets the stage for deriving the equation of a tangent plane to a surface at a given point, using the partial derivatives of the function and the coordinates of the point of tangency.

The paragraph details the process of finding the equation of a tangent plane to a surface at a specific point. It begins by considering the surface defined by a function Z=f(X,Y) and a point of tangency P with coordinates (X₀, Y₀, Z₀). The method involves holding one variable constant to define curves on the surface, calculating the slopes of these curves at point P, and using these slopes as direction vectors for the tangent plane. The normal vector to the tangent plane is found by crossing these direction vectors, and the equation of the tangent plane is then formulated using this normal vector and the point of tangency.

This section discusses the application of linear approximation and differentials to functions of single and multiple variables. It starts by reviewing the use of differentials (dy) as an approximation for changes in a function's value (ΔY) for single-variable functions. The concept is then extended to functions of two variables, where the tangent plane is used to approximate the change in Z (ΔZ), and the total differential (DZ) is introduced as an approximation tool. An example is provided to demonstrate the use of differentials in approximating the square root of a number, highlighting the accuracy of the approximation when the change in the input variable is small.

The paragraph extends the concept of linear approximation to functions of multiple variables. It explains how to use the tangent plane to approximate values on a surface near a given point. The linear approximation is expressed in different forms, emphasizing the components of the partial derivatives at the point of tangency. The paragraph also introduces the notation for the increment of Z (ΔZ) and the total differential (DZ), showing how they are used in approximations. An example is given to illustrate the process of approximating a function value using the total differential, which involves evaluating the function at a nearby point and adjusting based on the differential.

This paragraph provides an example of linearization and approximation for a function of two variables. It involves a function f(X,Y) and the process of finding its linearization at a specific point (a,b). The linearization is derived by evaluating the function and its partial derivatives at the point, and then using these to form the equation of the tangent plane, which serves as the linear approximation. The example concludes with using the linearization to approximate the function's value at a point close to the original point of linearization, demonstrating the application of the concept.

The paragraph explores the concept of differentiability for functions of two variables. It starts by recalling the definition of differentiability for single-variable functions and then presents the analogous definition for functions of two variables. The function is differentiable at a point if the change in the function's value (ΔZ) can be expressed as a sum of the partial derivatives times the changes in the variables (ΔX, ΔY), plus error terms that approach zero as ΔX and ΔY approach zero. The paragraph includes an example that demonstrates showing differentiability by manipulating the function's change to match the required form, and then introduces a theorem that simplifies the process by stating that if the partial derivatives exist and are continuous at a point, the function is differentiable at that point.

The final paragraph summarizes the lesson on differentiability and continuity for functions of two variables. It emphasizes the cumbersome nature of proving differentiability using the limit definition and introduces a theorem that simplifies the process by stating that the existence and continuity of partial derivatives at a point are sufficient for differentiability. The theorem is applied to the previously discussed function to demonstrate its use. The paragraph concludes the lesson with a confirmation of the function's differentiability over the entire domain of R², given the continuity of polynomials.